ABSTRACT: Eigenvalue decomposition and multiresolution are widely used techniques for signal representation. Both techniques divide a signal into an ordered set of components. The first component can be considered an approximation of the input signal; subsequent components improve the approximation. Principal component analysis selects components at the source resolution that are optimal for minimizing mean square error in reconstructing the original input. For classification, where discriminability among classes puts an added constraint on representations, PCA is no longer optimal. Features utilizing multiresolution have been demonstrated to preserve discriminability better than a single scale representation. Multiresolution chooses components to provide good representations of the input signal at several resolutions. The full set of components provides an exact reconstruction of the original signal. Principal component analysis with multiresolution combines the best properties of each technique: 1. PCA provides an adaptive basis for multiresolution. 2. Multiresolution provides localization to PCA. The first PCA-M component is a low-resolution approximation of the signal. Additional PCA-M components improve the signal approximation in a manner that optimizes the reconstruction of the original signal at full resolution. PCA-M can provide a complete or overcomplete basis to represent the original signal, and as such has advantages for classification because some of the multiresolution projections preserve discriminability better than full resolution representations.

Summary:

ABSTRACT (cont.): PCA-M can be conceptualized as PCA with localization, or as multiresolution with an adaptive basis. PCA-M retains many of the advantages, mathematical characteristics, algorithms and networks of PCA. PCA-M is tested using two approaches. The first approach is consistent with a widely-known eigenface decomposition. The second approach assumes ergodicity. PCA-M is applied to two image classification applications: face classification and synthetic aperture radar (SAR) detection. For face classification, PCA-M had an average error of under 2.5%, which compares favorably with other approaches. For synthetic aperture radar (SAR), direct comparisons were not available, but PCA-M performed better than the matched filter approach.

Due to the local nonstationarity of images, the autocorrelations differ by mode
(horizontal, vertical, diagonal) as well as by lag. The scatter matrix is doubly
symmetric but not (in general) Toeplitz. The first four eigenvectors are shown in
figure 4.8 and show that the filter is essentially separable along the horizontal and
vertical modes.

The weights (features) are separable even (low-pass) and odd (high-pass) decom-
positions. Figure 4.7( bottom-right) shows that the iterated filter bank develops
longer filters by cascading shorter filters. Short eigenfilter coefficients are driven by
PCA symmetry constraints and cannot adapt to data statistics.
Figure 4.9 (top) shows the outputs of three stages. For display purposes, each
image was normalized so that pixel intensities lie in the range (0, 1). The first
stage of the filter bank produces four outputs shown in the four top left panels of
figure 4.9. The four images are downsampled and arranged as a (2 x 2) array of
compressed images as shown in the top right panel. There is no implied ordering

or spatial relationship in the arrangement of the compressed images. We place the

compressed image to be iterated in the top-left of the (2 x 2) array. The low-pass

component (top-left) is passed to another iteration. The downsampled outputs of

the second stage are shown in the bottom, far left panel. The panel is di-p'l 't, ,I at

double scale. The third stage outputs are shown in the bottom, middle left panel

the figure 4.9. The (2 x 2) array of outputs from the third stage output is displ ,i'- .

at four times the actual scale. The outputs of all three stages are combined in the

are used. The output can be organized into 4 feature spaces that are 4 half-scale

images.

In each run, five training exemplars were randomly selected from the ten

exemplars available for each person in the ORL database. The results show great

sensitivity to selection of the training set. The sensitivity is not surprising. For

example, figure 5.7 shows a class that had the entire training set at one scale, and

the entire test set at another scale.

Training and Test data at different Scales

TRAIN

VERIFY

Figure 5.7: Training and Test Data at Different Scales

Giles et al. (1997) points out that a random selection among 40 classes would be

expected to be correct 1/40 = 2.5'. of the time. We feel that a more realistic base-

line for error rates is the performance of a template classifier with the raw data.

Since PCA is just a rotation, the performance would be the same as the perfor-
mance using all 200 eigenfaces. Samaria (1994) reported 10.5'. for the ORL data,
but we found that the error rate was also sensitive to training set and averaged
around 14.!' (first line of table 5.3). Some of the increased misclassification could
be due to the clipped data, but it is more likely that the decoupling of data due to
our classifier structure is responsible for the deterioration. The results seem to sup-
port that data organized into 4 independent feature spaces. The (2 x 2) window is
worse than data taken as a whole (raw data), but better than 16 decoupled feature
spaces. Note that if the feature spaces are linearly combined before classification,
we would expect an error rate similar to the raw images. All individual feature
classifiers and the iii ii i ly vote mechanism have a nonlinear operation when the
maximum output is selected.

5.5.4 Haar Multiresolution
A fixed Haar basis was used to crate a four level differential image pyramid. A
sample decomposition is shown along with the original image.
PCA-M Decomposition

I

U.

ORIGINAL RAW NORMALIZED

Figure 5.8: PCA-M Decomposition of One Picture

The autocorrelation matrix of the observation windows, as expected, shows that
the pixels in a natural image are 1/f. Classification using a Haar basis was not
significantly different from PCA-M since the Haar basis is well suited for 1/f
signals. Moreover, for small observation windows, the choice of multiresolution
basis is not very important. Multiplication by any fixed basis is a rotation; if all

the features are used, then input distances are preserved. Classification (with a

linear classifier) will be no better than using the raw inputs.

5.5.5 PCA-M

PCA-M was used to decompose images into multiresolution feature spaces

(components). Four feature spaces are 1/16 scale images, three are 1/8 scale

images, three are 1/4 scale images, and three are 1/2 scale images (figure 5.9).

Figure 5.9: Selected Resolutions

The decomposition was chosen to facilitate comparison to the Haar decomposition.

Referring to figure 5.9, components 1 to 4 have the longest eigenvectors; that is,